I received quite a few replies to my query aboutwhether or not a "random" polynomial over Q

(1) is irreducible in Q[x],

(2) has Galois group S_n over Q.

Although there has been some posted discussion ofwhat is meant by random, I haven't seen thefollowing references. (I don't know if theywere posted and have not appeared yet, or ifthey were just sent to me.) Anyhow here they areif anyone is interested. [Note to moderator:It would be nice if all postings were placedon some website where one could go to find a complete and up-to-date archive.]--------------------------------------------------Douglas Zare pointed out that (2) => (1).So if one is only interested in the end resultit suffices to establish (2). He also sketcheda proof of (2). --------------------------------------------------The oldest reference mentioned was supplied byWladyslaw Narkiewicz. He writes that both wereproved by B.L.van der Waerden in MathematischeAnnalen, 109, 1931, page 13.--------------------------------------------------Dani Berend sends the following also by vander Waerden:\item{[vW]} B. L. van der Waerden,Die Seltenheit der Gleichungen mit Affekt,{\it Math. Ann.} {\bf 109} (1933), 13--16.--------------------------------------------------Igor Shparlinski furnishes the following morerecent references on the subject:

See the beginning of J.-P. Serre, Topics in Galois theory. Another paper that might interest you is by Davis et al., ProbabilisticGalois theory of reciprocal polynomials, Expositiones Mathematicae16 (1998), 263-270. On a slightly different note there is the classic result of Schur thatif you cut off the Taylor series for e^x at x^n, then the resultingpolynomial over Q has Galois group S_n. For more recent work in thisdirection there are some papers of M. Filaseta; 97g:11025 and 97b:11034.--------------------------------------------------

Thanks to all who responded!

--Edwin Clark

------------------------------------------------------ W. Edwin Clark Department of Mathematics, University of South Floridahttp://www.math.usf.edu/~eclark/------------------------------------------------------